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These statements are commonly referred to as Thales' Theorem and the converse of Thales' Theorem.
Consider a right triangle ABC, with m∠B=90∘, inscribed in a circle.
According to the Inscribed Angle Theorem, the measure of ∠B is half the measure of its intercepted arc, AC.
Substituting m∠B=90∘ into the equation above gives that mAC=180∘. Then, AC is a semicircle, implying that AC (the hypotenuse of △ABC) is a diameter of the circle.
Consider a triangle ABC inscribed in a circle such that one side of the triangle is a diameter of the circle.
Since AC is a diameter, then AC is a semicircle and then mAC=180∘. Now, the Inscribed Angle Theorem gives a relation between this arc and the angle opposite to the diameter.